CERTAIN IDENTITIES OF ZETA FUNCTIONS ON A QUATERNION ALGEBRA

• Journal title : Honam Mathematical Journal
• Volume 35, Issue 1,  2013, pp.73-82
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2013.35.1.73
Title & Authors
CERTAIN IDENTITIES OF ZETA FUNCTIONS ON A QUATERNION ALGEBRA
Kim, Insuk; Jun, Sungtae;

Abstract
There are some identities between the representations of Hecke operators defined on a subspace of modular forms associated with orders of a Quaternion algebra. In this paper, we find the identities between the zeta functions attached to these representations.
Keywords
zeta function;order;quaternion algebra;hecke ring;
Language
English
Cited by
References
1.
M. Eichler, The basis problem for modular forms and the traces of Hecke operators. Springer-Verlag, Lecture Notes in Math., Vol.320, pp. 75-151, 1972.

2.
R. Godement, Les fonctions ${\zeta}$ des algebras simples I, II, Seminaire Bourbaki, 1958/1959.

3.
H. Hijikata, A. Pizer, T. Shemanske, The basis problem for modular forms on ${\Gamma}_0(N)$ Memoirs AMS, Vol.32 No.418, November, 1989.

4.
S. Jun, The equalities between Brandt matrices on primitive orders, preprint.

5.
S. Jun, On the representatives of modular forms on ${\Gamma}_0(N)$, J. Algebra, Vol.64, pp. 340-390, 1980.

6.
A. Pizer, An algorithm for computing modular forms by theta series, J. KMS, Vol.34 No.4, pp. 808-816, 1997.

7.
H. Shimizu, On the traces of Hecke operators, J. Fac. Sci. Univ. Tokyo, Vol.10, pp. 1-19, 1963.

8.
H. Shimizu, On zeta functions of quaternion algebra, Ann. of Math., Vol.81, pp. 166-193, 1965.

9.
G. Shimura, "Introduction to the arithmetic theory of automorphic functions", Princeton Univ Press, Princeton, N.J., 1971.

10.
T. Tamagawa, On ${\zeta}$ functions of a division algebra, Ann. of Math. Vol.77, pp. 387-405, 1963.